§Provide our pre-service teachers with a capstone course that
makes connections between and among the ideas covered in upper level math
courses to topics taught in secondary school. Core topics of a typical
high school math program are presented from an advanced perspective, revealing
their relationships to college level math. This perspective and analysis
of topics provides teachers with a deeper understanding of high school math and
fosters an appreciation of its structures and beauty.

§Provide a course for in-service teachers that better
prepares them to handle the challenges of developing teaching strategies and
assessment tools for the secondary school standards.

§Better prepare pre-service teachers for the newly proposed
licensure exams and provide a mechanism for making improvements once pass rates
on state exams become available.

Course Objectives and Assessment:

The NCTM has set standards for the
mathematics preparation of all mathematics teacher candidates. This
course addresses several of these standards.

1.Knowledge of Problem Solving.
In this course, candidates demonstrate that they know, understand and apply the
process of mathematical problem solving by successfully completing the problem
set for the course.

2.Knowledge of Reasoning and Proof.
Candidates get further experience with reasoning, constructing, and evaluating
mathematical arguments in this course.

3.Knowledge of Mathematical Communication.
This course is structured to allow many opportunities for the candidates to
communicate their ideas, orally and in writing. Assignments include
classroom discussion, presentations, and papers.

4.Knowledge of Mathematical Connections.
The main emphasis of this course is to give candidates a structure in which to
make connections between different mathematical ideas and build a more complex
mathematical understanding. A class project will be completed to
demonstrate an understanding of such a connection.

5.Knowledge of Mathematical Representation.
The main representations are verbal, graphical, numerical, and analytic.
Each type of representation is included in the problem list for the
course. Emphasis will be given to proper use of math symbols and
terms.

6.Knowledge of Technology. The
Mathematics Content Knowledge exam that will be required for teacher candidates
entering the work force requires the use of graphing calculators. In this
course we will use a graphing calculator as often as possible to further enhance
the solution and illustration of problems.

7.Dispositions. An appreciation
of mathematics often comes after much knowledge is acquired and has become
intuitive. When connections are made, moments of greater understanding
are reached, “Aha” moments. There is a great deal of satisfaction in this
understanding that is often hard sought. We make the connections in
this course and hope that the chances that the teachers themselves become
lifelong learners are increased.

8.Knowledge of Mathematics Pedagogy.
This course will serve to deepen the understanding of how students learn
mathematics, by providing examples of questions given to students in the
secondary level courses and seeing what concepts are being addressed.
In-service and pre-service teachers will be given assignments to design
questions that address the items given in the Grade Span Expectations (GSEs) and Grade Level Expectations (GLEs).

9.through 15. The content areas of knowledge for
secondary level mathematics teachers are provided in their other courses, with
the exception of Discrete Mathematics. In this course, we review
many of those ideas as they come up in examples. Teacher candidates need
to have knowledge of the math that students will encounter when they leave high
school, either in college, vocational school, or on the job.

Exercises: You will be given a problem list from the
textbook to work on at home and during class. These problems provide a
mechanism for reviewing advanced math and making connections among mathematical
ideas.

Reports: Two written assignments will be given, using
sources other than our text to address issues related to the teaching of
mathematics. Discussion of the reports will be held in class.

World Wide Web Assignment: Two WWW assignments will be given that will
help teachers to explore the usefulness of the web for teaching
mathematics. They will be sent by the instructor to the class members by
e-mail at the beginning of the course and the completed work is to be sent back
to the instructor by e-mail by the last day of class.

Exams: Short midterm and final exams will be
given. The questions are similar in scope to those that might be given on
a licensure exam for beginning secondary mathematics teachers.

Projects:
From a list of topics, Investigate and analyze the topic and explore
connections between and among mathematical concepts and applications.
These are to be presented to the class. A handout will be given to each
class member and a final report will be given to the instructor. Students
may work in groups, but each class member is responsible for an entire project
including the report and presentation. Design two GSE or GLE questions
related to your topic. You are required to complete a 15-20-minute
midterm and a 25-30-minute final project (No longer!)

If you have a documented
disability, which may require individual accommodations, please make an
appointment with me prior to the class meeting. We will discuss how to meet
your needs to ensure your full participation and fair assessment procedures.

Reports: Pick a question
from this list, consult at least one source other than our book (see material
in class), and write a 2-3 page report. All quotes are to be properly
cited and your source material needs to be well documented.

Topics for Reports:

1.There are
many issues related to the topic of cheating. How can we prevent it in the
classroom? How do we deal with it when it happens? How do you feel
about cheating?

2.Besides knowledge of topics in
mathematics, what outcomes do we want our students to achieve as a result of
their high school education? Ideas: properly using and
understanding symbolism, having an open mind to new ideas, be able to
understand multiple forms of a correct answer and alternative equivalent
definitions, be motivated to continue in mathematics, basic mathematical
knowledge becomes part of the students intuition, understanding that
mathematics is a system of statements built on logic, be able to apply
deductive reasoning, understanding that mathematical statements are either true
or false.

3.Do students want to know
WHY? Take, for example, the quadratic formula. We want high
school students to learn to apply it but also understand why it is true.
Take the formula for the derivative of a power function in calculus. Can
you explain why it is true? Is it important to you to know why?

4.College level students are often
weak in using fraction, exponents, logarithms, and trig functions. Drill
in these areas can start by into using specific numbers and build to simplifying
examples using formulas with variables and constants.

5.Can anyone learn math? What
issues are common among students who say they are not good at math, or not math
people? It takes effort to learn the subject. Some students have no
motivation to do so. Can we change their minds? Do we need to
convince more students to continue to study math in college? Find some
articles.

6.What questions could a high
school student ask that may require college level understanding of mathematics
to answer. Give examples of questions and answers that are broken down to
the high school level.

7.What is abstract thinking?
Give examples. Compare to metaphors.

8.What is the difference between forms of functions, types of functions and families of functions? Give examples of each.

9.As teachers, you have to be good
at math. What does that mean?

10.How can you find time for each individual student to find his or her
weaknesses?

11.What are some of the major stumbling blocks from elementary
education? Fractions!

12.Use the GSEs and GLEs
for this question. Choose one goal. What is the best way to learn
this concept or topic? What learning materials can you use to teach this
concept or topic?

13.What are some ideas for courses that are not generally taught in high
school? How would you fit this course into the high school
curriculum?

14.The development of a mathematical idea may have been initiated by an
application. Name some mathematical concepts and their
applications. For instance, parallel.
Using the definition: two curves in 2-dimensional space are parallel if they
are equidistant apart, train tracks are parallel. What about lines of
longitude and latitude?

15.The more skilled and practiced you are at mathematics, the more of it is
in your subconscious. You have more intuition. This happens over
time. Your subconscious is always at work. How can you use this to
problem solve? Can you do this when you are experiencing math anxiety?

16.Can we teach abstract thinking? Give examples from math and real
life that may illustrate abstract thinking. Example from math the number
system of the real numbers under addition and the positive real numbers under
multiplication are both groups.

17.How is deductive reasoning applied in other subjects outside of math?

18.Name some words that mean something in English and are co-opted to mean
something else in math. Name some words that have no meaning outside of
math.

19.What is a number? Is a telephone number a number? What is a number
system?

20. I have heard students say, “I hate proofs”, even teacher
candidates. What is behind this statement? How does learning to
prove theorems help teacher candidates to be better teacher?

21.Other…

Ideas for Projects

1.Given a function f(x) and a number a in the domain of f(x),
find the formula for a
polynomial P(x) of degree 3 that has the following properties.The first, second, and third derivatives of f(x)
and P(x) at x=a are the same.Compare your answer to the Taylor Polynomial of degree 3 for f(x) at x=a.(This question leads to many others and can be turned into a project.)

2.There is a division theorem about the
integers that says, for any 2 integers, a and b, when dividing a by b, there
is a remainder r, which is an integer such that 0 ≤ r<b.Similarly, for polynomials, there is a theorem that says, for any 2
polynomials, p(x) and g(x), when
dividing p(x) by g(x), there
is a remainder r(x), which is a polynomial that such that the degree
of r(x) is less than that of g(x).Explain what Integers have in common with polynomials that would explain
this similarity.

3.Applications often drive the development of
new mathematics.What problems were
being worked on that led to the development of the derivative and integral?

4.Other questions involving applications.

5.Give examples of symbols that are used to
mean different things.

6.Consider the usual definition and
characterization theorem of combinatorial trees that gives 4 alternative
definitions.Under what circumstances
would one of the alternative definitions be more intuitive than the usual
definition?

7.You learned to derive the volume of a cone in
Calculus.Can you use high school math
to explain the formula for the volume of a cone?

8.We often try to sidestep the rigorous
definition of the limit of a function of x as x approaches infinity by using a more intuitive
explanation.Give a description of the
meaning of the limit using pictures that is useful and accurate.

9.Use the epsilon-delta definition of the limit
to prove that limit x -> 2
of x^2=4.

10.Project - Factoring polynomials.What did you learn from calculus and abstract
algebra about factoring of polynomials?What type of polynomials can we ask high school students to factor (quadratics,
cubic, higher powers)?What type would
they have no chance to factor and why?Make a lesson that explores using the graphing calculator to graph and
find roots of polynomials.What does the
graph say about the roots being real or imaginary numbers? Break down into
elemental pieces and drill.

11.Write a lesson that teaches using the
quadratic formula to solve quadratics.Apply to examples like:a x^2
+ a x
+ b.Break down into elemental pieces and drill.

12.Write a lesson that teaches the difference
between variables and constants.Break
down into elemental pieces and drill.

13.Project - In your college courses, you
learned the definition of a linear function.Is a
general line a linear function?What
functions are linear?Give examples of some
that are and some that aren’t.Prove
your answers.Relate this to shifting,
compressing, and stretching of graphs of functions.Write a series of lessons for high school
students on this topic.Break down into
elemental pieces and drill.

14.Define even and odd functions.For each type, give examples of functions
that have the property and some that don’t.Can a function be both even and odd?For even and odd functions, what properties do their respective Taylor
Series have?

15.Make up some examples of problems for high
school students that can be solved using arithmetic or algebra.Devise extensions of these problems that
require algebra.For example:Jane has an average of 87 after 4 tests.What score does she need on the fifth test to
average 90 for all 5?If the test grade
is at least 0 and at most 100, what are her possible overall averages?Draw a graph.

16.Some questions that they have probably never
seen in any college course.How to
solve?Brainstorm, free flow of ideas.

17.Solve all questions in the sample Content
Knowledge test in Mathematics for Secondary School Teachers.Relate each question to the college course in
which you learned the topic.

18.Explain the relationship between real,
complex, and imaginary numbers in the context of college abstract algebra.Give examples of each.Find a way to motivate learning about complex
numbers to high school students.

19.Illustrate an example given in one of the
books:Proofs
Without Words, and Proofs
Without Words II.

22.Real numbers can be defined in two ways.Describe the top down approach of using
complete ordered fields and the bottom up approach using rational numbers.

23.Show that the square root of [ 3 + the square
root of 7 – the square root of (8 – 2 times the square root of 7) ]is rational.I haven’t solved this yet!

24.Model the following situation with an
equation….. There are many examples of this type.

25.Make this into many exercises:Solve some challenging problems in Geometry,
involving proofs.There are many to
select from in the following book list.All of these are being ordered for the library. Methods of Geometry,
by Smith; A History of Non-Euclidean Geometry: Evolution of the Concept of a
Geometric Space, by Rosenfeld, Grant, and Shenitzer;
Challenging Problems in Geometry,by Posamentier, and Salkind;
Symmetry, Shape and Space, by Kinsey and Moore; Active Geometry,
by Thomas; Transformation Geometry: An Introduction to Symmetry, by
Martin;Elementary Mathematics from
an Advanced Standpoint: Geometry, by Klein, Hedrick, and Noble;Euclidean and Non-Euclidean Geometry:Development and History, by Greenberg.